Fichier PDF

Partage, hébergement, conversion et archivage facile de documents au format PDF

Partager un fichier Mes fichiers Convertir un fichier Boite à outils Recherche Aide Contact



Anthropological Theory 2010 Bachir Diagne 62 6 .pdf



Nom original: Anthropological Theory-2010-Bachir Diagne-62-6.pdf
Titre: 156x234mm
Auteur: Allsetjb

Ce document au format PDF 1.4 a été généré par QuarkXPress(R) 8.1 / Acrobat Distiller 9.3.2 (Macintosh), et a été envoyé sur fichier-pdf.fr le 28/03/2013 à 11:08, depuis l'adresse IP 64.134.x.x. La présente page de téléchargement du fichier a été vue 492 fois.
Taille du document: 240 Ko (6 pages).
Confidentialité: fichier public




Télécharger le fichier (PDF)









Aperçu du document


Anthropological
Theory
http://ant.sagepub.com/

0 and 1 in our digital age
Souleymane Bachir Diagne
Anthropological Theory 2010 10: 62
DOI: 10.1177/1463499610365356
The online version of this article can be found at:
http://ant.sagepub.com/content/10/1-2/62

Published by:
http://www.sagepublications.com

Additional services and information for Anthropological Theory can be found at:
Email Alerts: http://ant.sagepub.com/cgi/alerts
Subscriptions: http://ant.sagepub.com/subscriptions
Reprints: http://www.sagepub.com/journalsReprints.nav
Permissions: http://www.sagepub.com/journalsPermissions.nav

>> Version of Record - Jun 7, 2010
What is This?

Downloaded from ant.sagepub.com by guest on March 28, 2013

Anthropological Theory
Copyright © 2010 SAGE Publications
(Los Angeles, London, New Delhi,
Singapore and Washington DC)
http://ant.sagepub.com
Vol 10(1–2): 62–66
10.1177/1463499610365356

0 and 1 in our digital age
Souleymane Bachir Diagne
Columbia University, USA

Abstract
This paper examines the history of algebraic logic as developed by German
philosopher G.W. Leibniz (1646–1716) and British mathematician George Boole
(1814–64). It shows the role and significance of the two numbers 0 and 1 in the
newly created algebra of logic and how that explains why those numbers have come to
be the very symbols of our digital age.
Key Words
algebra • Boole • De Morgan • digital • equation • interpretation • Leibniz • logic •
quantification • symbolism

WHAT KIND OF NUMBERS ARE 0 AND 1?

If we were asked to pick up a couple of numbers that symbolically express our times, no
doubt that the best choice would be 0 and 1. We do not need to be mathematicians to
know that these two symbols are used in sequences such as 00110011 . . . to encode
everything and thus truly characterize our civilization of the computer. This contribution visits the philosophy and the history behind the presence and significance of these
two numbers in this digital age of ours.
Let us first ask the following question: what kind of numbers are 0 and 1? Are they
really numbers of the same kind as the others: 3, 4, etc.? To answer that question,
consider the following characteristic property of the two symbols. We know that for all
numbers x, x2 is greater than x, except in the case of 0 and 1 since we have:
0 ⫻ 0 = 02 = 0
and
1 ⫻ 1 = 12 = 1
The property a*a = a where * is a given operation is known as the idempotent law. So if
we consider the set of only two numbers {0,1}, the equations above express the
idempotent law for the operation x. This property sets 0 and 1 apart from the other
numbers as being non-quantitative, so to say. Or to speak with more precision, when
we restrict ourselves to a set containing 0 and 1 as its only elements, the idempotent law
62

Downloaded from ant.sagepub.com by guest on March 28, 2013

DIAGNE 0 and 1 in our digital age

x2 = x
is an axiom in that non-numerical or non-quantitative type of algebra with which we
are then dealing.
The possibility of creating such a non-quantitative algebra restricted to those two
numbers is precisely the reason for the crucial role they have played in the emergence of
the field known as the ‘algebra of logic’ and, subsequently, in our civilization of the
computer. Algebraic logic was created – first envisioned by the philosopher G.W. Leibniz
(1646–1716), it was fully realized by the British mathematician George Boole
(1815–64) – as a particular case of non-numerical algebra. That is an algebra where
literal symbols do not necessarily stand for quantities but could represent, say, concepts
such as men, pebbles, lions, etc. Thus if ‘a’ represents ‘lions’ and the operation x stands
for ‘and’ (also represented as ≅ in set theory), we will understand a2 = a as meaning
simply: ‘lions that are lions are simply lions’. As George Boole, who first fully organized
the encounter between algebra and logic, thus creating the field of algebraic logic,
famously wrote: it is not of the essence of mathematics to be conversant with the notions
of number and quantity.
ORGANIZING THE ENCOUNTER BETWEEN ALGEBRA AND LOGIC

This encounter between the language of algebra and classical Aristotelian logic became
possible when two revolutions happened, the first one in logic, the second in mathematics.
(1) In logic, a crucial reform happened that allowed the transformation of the canonical form of the logical sentence ‘S is P’ (where S is the subject, P the predicate and ‘is’
the copula) into a logical equation where the copula is replaced by the symbol of equality
=. That reform was the quantification of the predicate and is to be attributed to the
logician Augustus De Morgan (1806–71). What does that reform mean? It is only
possible to put a sign of equality between the subject and the predicate when those two
terms have the same ‘quantity’. One cannot write all men = mortal because there are far
more mortals than just men. In other words, the quantity of mortals exceeds that of
men. And that leads to the realization that when one says ‘all men are mortal’ one really
means and understands that ‘all men are some mortal’. If that were explicitly stated, then
it would be possible to get rid of the copula and simply write, using the language of
equations: all men = some mortal. What Augustus de Morgan discovered, in summary,
is precisely that by quantifying the predicate also (and not just the subject), it is possible
to write logical sentences under the form of equations and treat them as such according
to the rules defined in the science of equations, which is algebra.
(2) The revolution that happened in mathematics has already been evoked: algebra
was ‘de-quantified’ when it was established that the literal symbols do not necessarily
have to represent numbers or quantity. Quantity is just one interpretation of the symbolism of algebra. The possibility of non-numerical algebras meant that one could create
an algebraic system in which our reasoning could be expressed. Such an algebra is what
George Boole invented, retrieving many of Leibniz’s intuitions without knowing them
since most of the philosopher’s works in the field of logic remained unpublished until
their discovery, in the early 20th century, by Louis Couturat. What does that system
look like?

63

Downloaded from ant.sagepub.com by guest on March 28, 2013

ANTHROPOLOGICAL THEORY 10(1–2)

THE ELEMENTS OF BOOLE’S ALGEBRA OF LOGIC

In the system designed by George Boole:






concepts or classes of things are represented by literal symbols: x, y, z, etc. One
particular symbol, v, represents the concept ‘some’;
there are two symbols of operation: x (the logical multiplication corresponding to
‘and’); + (the logical addition corresponding to ‘or’); the inverse of logical addition
is naturally represented by – (minus);
the symbol of equality is written =;
the symbol 1 represents the totality, or the ‘universe’, and 0 represents ‘nothing’, the
empty class.

Let us express, for example, the four logical propositions forming what is traditionally referred to as the logical square. Those are the universal affirmative (A) ‘all men are
mortal’, the universal negative (E) ‘no man is mortal’, the particular affirmative (I) ‘some
men are mortal’ and the particular negative (O) ‘some men are not mortal’. When x
represents the class of ‘men’, y that of mortal beings (v being the indeterminate class of
some things), those four propositions are translated in symbols as follows:
A: x = vy
That equation can be proven to be equivalent to x(l–y) = 0 (All men are mortal means
that men that are not mortal, that is 1–y, do not exist).
E: xy = 0 (No man is mortal means that mortal men do not exist);
I: xy = v (Some men are mortal means that men that are mortal are a certain –
indeterminate – class of beings);
O: x(l–y) = v (Some men are not mortal means that men that are not mortal are a certain
– indeterminate – class of beings).
The importance of x2 = x appears: all operations on equations in which logical
propositions are translated will be carried according to the usual rules of algebra with
the restriction that the ‘index law’ x2 = x is the governing rule of those transformations.
Boole calls it ‘the fundamental law of the system’.
LOGICAL DEDUCTION AS ALGEBRAIC TRANSFORMATIONS

The purpose of designing an algebraic system of logical reasoning is as follows. Given
certain data (premises), one can translate them into the algebraic symbolism, then treat
the equations thus obtained through the usual algebraic machinery while obeying the
index law; at the end of the procedures the results of the transformations will be retranslated into our natural language.
Let us take the simple case of the index law: The equation x2 = x (translating, for
example, ‘men that are men are men’) can be transformed into x2 – x = 0; from it follows
that x(1–x) = 0 (a simple algebraic transformation). Now, how do we translate this
algebraic result into natural language? Obviously, since 1–x means the non-Xs, the
formula reads: ‘men that are not men do not exist’. In other words, this is the expression
of the fundamental law of thought known as the principle of contradiction.
64

Downloaded from ant.sagepub.com by guest on March 28, 2013

DIAGNE 0 and 1 in our digital age

The algebraic procedures may be far more complex and even go through steps that
seem deprived of any logical significance. That is a remarkable feature of Boole’s system
in that he is confident that, in the end, the result produced mechanically (Leibniz would
say ‘blindly’) by the algebraic procedures will always find a logical interpretation in our
natural language.
Let us consider one such complex example. Given that:
z = men
x = rational
y = animal
we pose that:
Men are rational animals: z = xy.
And we ask the question:
What kind of objects are the ‘rational things’, x?
From z = xy it comes, algebraically: x = z/y. This is a formula with no logical meaning.
We need therefore a ‘development’ of it. What is that? When we have a function of one
class, x, that we write f(x), we can develop it into the form
f(x) = Ax + B(1–x).
We see immediately that f(1) = A and f(0) = B. So the development of f(x) is
f(1)x + f(0) (1–x)
In a similar way the development of f(x,y) is
f(1,1) xy + f(1,0) x(1–y) + f(0, 1) (1–x)y + f(0,0) (1–x)(1–y)
Applying that expansion to our case x = z/y, we proceed with the development and get
x = lyz + 0y (l–z) + 1/0 (1–y) z + 0/0 (1–y)(1–z)
In order to produce a logical interpretation of the result we must consider the following
translations:
1:

it means ‘the universe’; so we have to take the totality of the class to which the
symbol is prefixed.
0: it means ‘nothing’; so we do not consider the class to which the symbol is prefixed.
0/0: since in algebra it is the symbol for the indeterminate, it is here equivalent to the
symbol of indeterminate class v and means ‘some’; so we take an indeterminate
portion of the class to which the symbol is prefixed.
65

Downloaded from ant.sagepub.com by guest on March 28, 2013

ANTHROPOLOGICAL THEORY 10(1–2)

1/0: it is a symbol for the impossible; so we affirm the inexistence of the class to which
the symbol is prefixed.
As a result, the logical interpretation of the development of x is: Rational beings are all
animals that are men and an indeterminate portion of beings that are not animals and that
are not men; and it is also affirmed that men that are not animals do not exist.
These examples show what it means for us to have realized our intellectual procedures
in the machinery of algebra (and in physical machines). It means that we have thus
created an objective equivalent of our intelligence: when we put in data at one end it
spits out conclusions at the other end, blindly, mechanically.
CONCLUSION

Is there more in our human intelligence than what we have transferred to the machine?
Is there more in our reality than the world of the matrix, endlessly generated by
sequences of 0’s and 1’s? These are examples of the many questions raised by the
ubiquitous presence of these special numbers 0 and 1. In fact these are not ‘numbers’
any more. As symbols for [the totality] and for [nothingness] they are the cornerstone
of the very structure of the human mind and of human reasoning, as Boole argues in
his main work published in 1854 under the title An Investigation of the Laws of Thought
on Which Are Founded the Mathematical Theories of Logic and Probabilities.
Those who first came to realize that the procedures of human thought could be translated into the language of algebra and thus made mechanical can be considered the
creators of our civilization of the computer. Leibniz was certainly the first, even if his
work remained unpublished and unknown. Boole made real Leibniz’s dream of transforming reasoning into a calculus. Both thought that with algebra human reasoning had
found the original Adamic language spoken by humanity before the curse of Babel, and
that 0 and 1 encapsulated the power of the human mind to reason in the universal
language of algebra. There are certainly many ways in which our digital age, the age of
0 and 1, proves them right.
SOULEYMANE BACHIR DIAGNE is an alumnus of Ecole Normale Supérieure (rue d’Ulm) and holds a
PhD from Sorbonne University in Philosophy. He taught philosophy at Cheikh Anta Diop University, Dakar
(Senegal), for 20 years before joining, in 2002, Northwestern University as a Professor in the Departments of
Philosophy and Religion, with an affiliation with the Department of French and Italian. In 2008 he joined
Columbia University as a Professor in the Department of French and Romance Philology and in the Department of Philosophy. His areas of research and publication include the history of philosophy, the history of
logic and mathematics, Islamic philosophy, African philosophy and Francophone literature. Address: Department of French and Romance Philology, Columbia University, Philosophy Hall, mc 4902, 1150 Amsterdam
Ave., New York, NY 10027, USA. [email: sd2456@columbia.edu]

66

Downloaded from ant.sagepub.com by guest on March 28, 2013


Documents similaires


Fichier PDF anthropological theory 2010 bachir diagne 62 6
Fichier PDF technologies du numerique ii fonctionnement de l ordinateur
Fichier PDF algebre booleenne esaber
Fichier PDF complex logic a zinoviev dordrecht 1973
Fichier PDF finkel10
Fichier PDF logical physics a zinoviev reidel 1983 600dpi lossy


Sur le même sujet..